# Master the Art of FTOC: Comprehensive Examples and Expert Advice

Recent questions in FTOC
grenivkah3z 2022-07-08

## What is the difference between first and second fundamental theorem of calculus?

Sam Hardin 2022-07-05

## Given a continuous function $f:\left[0,1\right]\to \mathbb{R}$, prove that$\mathrm{\forall }t>0,\frac{1}{t}\cdot \mathrm{ln}\left({\int }_{0}^{1}{e}^{-tf\left(x\right)}dx\right)\le -minf\left(x\right).$

kolutastmr 2022-07-02

## Finding $f\left(x\right)$ in ${\mathrm{cos}}^{2}\left(x\right)f\left(x\right)={x}^{2}-2{\int }_{1}^{x}\mathrm{sin}\left(t\right)\mathrm{cos}\left(t\right)f\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t$

Crystal Wheeler 2022-07-01

## How do you use the Fundamental Theorem of Calculus to find the derivative of $\int \frac{1}{1+{t}^{2}}dt$ from $x$ to $5$?

pipantasi4 2022-07-01

## Finding the maximum of $G\left(x\right)={\int }_{x}^{x+a}f\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$ using FTOC.

lilmoore11p8 2022-07-01

## Find the derivative of the following function using the Fundamental Theorem of Calculus:$F\left(x\right)={\int }_{{x}^{3}}^{{x}^{6}}\left(2t-1{\right)}^{3}dt$

glitinosim3 2022-07-01

## Let $f$ be continuous on $I=\left[a,b\right]$ and let $H:I\to \mathbb{R}$ be defined by $H\left(x\right)={\int }_{x}^{b}f\left(t\right)dt,x\in I$. Find ${H}^{\prime }\left(x\right)$.

Leland Morrow 2022-07-01

## Can you apply the fundamental theorem of calculus with the variable inside the integrand?For example: $\frac{d}{dx}\left({\int }_{a}^{x}xf\left(t\right)dt\right)$

Sarai Davenport 2022-06-28

## What is the difference between the two parts of FTOC?

oleifere45 2022-06-25

## How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions?

Emanuel Keith 2022-06-25

## $f\left(x\right)={\int }_{0}^{\mathrm{sin}x}1+\mathrm{sin}\left(\mathrm{sin}\left(t\right)\right)dt$Find $\left({f}^{-1}{\right)}^{\prime }\left(0\right)$

excluderho 2022-06-24

## Derivative of a function and an integral$\frac{d}{dx}\left({x}^{6}\left({\int }_{0}^{sinx}\sqrt{t}dt\right)\right)$

Yahir Tucker 2022-06-24

## We know if $g$ is continuous on $\left(a,b\right)$ and $F\left(x\right)={\int }_{a}^{x}g\left(t\right)dt$, then${F}^{\prime }\left(x\right)=g\left(x\right)$But, how about if we have$F\left(x\right)={\int }_{a}^{h\left(x\right)}g\left(t\right)dt$What should ${F}^{\prime }\left(x\right)$ be?? can we still apply fundamental theorem of calculus?

Petrovcic2x 2022-06-22

## Can use FToC to evaluate $\underset{x\to \mathrm{\infty }}{lim}\frac{{\int }_{0}^{x}\phantom{\rule{mediummathspace}{0ex}}f\left(t\right)dt}{{x}^{2}}$?

Poftethef9t 2022-06-20

## If $f\left(x\right)$ is even, then what can we say about:${\int }_{-2}^{2}f\left(x\right)dx$If $f\left(x\right)$ is odd, then what can we say about${\int }_{-2}^{2}f\left(x\right)dx$Are they both zero? For the first one if its even wouldn't this be the same as${\int }_{a}^{a}f\left(x\right)dx=0$Now if its odd $f\left(-x\right)=-f\left(x\right)$. Would FTOC make this zero as well?

watch5826c 2022-06-19

## Show that $g$ is differentiable and find ${g}^{\prime }\left(x\right)$, FTOC

Carolyn Beck 2022-06-17

## Find the derivative of integral$H\left(x\right)={\int }_{3}^{{\int }_{1}^{x}{\mathrm{sin}}^{3}tdt}\frac{dt}{1+{t}^{2}+{\mathrm{sin}}^{6}t}$

juanberrio8a 2022-06-15

## Prove ${\int }_{-a}^{a}f\left(x\right)dx=0$assuming $f\left(x\right)$ is odd.

Poftethef9t 2022-06-11

## Why does FTOC apply here, to find the derivative of ${\int }_{\mathrm{sin}\left(x\right)}^{\pi }\frac{t}{\mathrm{cos}\left(t\right)}dt$

anginih86 2022-06-08

## Suppose $F\left(x\right)={\int }_{3x+8}^{{x}^{2}+5x+1}{\mathrm{csc}}^{2}\left(t\right)dt$. How would one find ${F}^{\prime }\left(x\right)$ using the first fundamental theorem of calculus?

It's hard to find a subject as complex as the integral calculus and the related equations that are used to solve the calculus theorem problems. One of them is a fundamental theorem of calculus, which is essential for a correct understanding of the topic. Looking at the questions and answers below, you can receive free fundamental theorem of calculus help based on what is being asked. We also provide both short and detailed fundamental theorem of calculus examples and solutions that will be similar to your college task. Feel free to explore more than one solution for the best results!